Reference no: EM132190200
Factorial Design at Two Levels and Response Surface Method
Question 1: Factorial design at two levels.
An experiment was conducted on an operating chemical process in which four factors were studied in a 24 factorial design. Shown in the table below are the factor levels, the design, the random order in which the runs were made, and the response (impurity) at each of the 16 reaction conditions:
Factor Levels
|
1
|
Concentration of Catalyst (%)
|
5
|
7
|
|
2
|
Concentration of NaOH(%)
|
40
|
45
|
|
3
|
Agitation speed (rpm)
|
10
|
20
|
|
4
|
Temperature (F)
|
150
|
180
|
Coded Factor Levels of the Experiment Design
|
Run Order
|
1
|
2
|
3
|
4
|
Impurity
|
|
2
|
-
|
-
|
-
|
-
|
0.38
|
|
6
|
+
|
-
|
-
|
-
|
0.4
|
|
12
|
-
|
+
|
-
|
-
|
0.27
|
|
4
|
+
|
+
|
-
|
-
|
0.3
|
|
1
|
-
|
-
|
+
|
-
|
0.58
|
|
7
|
+
|
-
|
+
|
-
|
0.56
|
|
14
|
-
|
+
|
+
|
-
|
0.3
|
|
3
|
+
|
+
|
+
|
-
|
0.32
|
|
8
|
-
|
-
|
-
|
+
|
0.59
|
|
10
|
+
|
-
|
-
|
+
|
0.62
|
|
15
|
-
|
+
|
-
|
+
|
0.53
|
|
11
|
+
|
+
|
-
|
+
|
0.5
|
|
16
|
-
|
-
|
+
|
+
|
0.79
|
|
9
|
+
|
-
|
+
|
+
|
0.75
|
|
5
|
-
|
+
|
+
|
+
|
0.53
|
|
13
|
+
|
+
|
+
|
+
|
0.54
|
(a) Make the table of contrast (a table of plus and minus signs from which all the main effects and interaction effects be calculated).
(b) Calculate all the main and interaction effects. Show the steps of your calculation and only run software to check your answer.
(c) Assuming the three-factor and higher order interactions to be noise, compute an estimate of the error variance of the effects. Show the steps of your calculation and only run software to check your answer.
(d) Based on your results from Question 1(b) and (c), which of the estimated effects are likely to be the real effects rather than noise? Why? How would you interpret each of the real effect(s)? Show your interpretation graphically.
(e) Assume that the present conditions of operation are 1=5%, 2=40%, 3=10 rpm, 4=180F. Make recommendations of better operation conditions to reduce the impurity.
(f) Assume your conclusions from Question 1(d) remain valid for operation conditions beyond the range enclosed by the two specified levels of the four factors. If you are able to conduct a few more experiment runs, describe how you would set and/or vary the four factors in the additional experiments so that, with as few experiments as possible,
i. You may confirm/ further investigate each of the real effects found in Question 1(d), and
ii. You could most likely find even better operation conditions than your answer to Question 1(e).
Question 2: Estimate the error variance of the effects.
A chemist used a 24 factorial design to study the effects of temperature, PH, concentration, and agitation rate on yield (measured in grams), without replicated runs. If we know the standard deviation of an individual observation is 6 grams, what is the variance of the temperature effect? Hint: use the basic properties of variances of random variables.
Question 3: Application of response surface method.
The data table below came from a tire radial run-out study. Lower run-out values are desirable.
The factors under study were:
• x1 = Post-inflation time (minutes),
• x2 = Post-inflation pressure (psi),
• x3 = Cure temperature (0F),
• x4 = Rim size (inches), and
The response variable was:
• y = Radial run-out (mils).
|
|
x1
|
x2
|
x3
|
x4
|
y
|
|
1
|
9.6
|
40
|
305
|
4.5
|
0.51
|
|
2
|
32.5
|
40
|
305
|
4.5
|
0.28
|
|
3
|
9.6
|
70
|
305
|
4.5
|
0.65
|
|
4
|
32.5
|
70
|
305
|
4.5
|
0.51
|
|
5
|
9.6
|
40
|
335
|
4.5
|
0.24
|
|
6
|
32.5
|
40
|
335
|
4.5
|
0.38
|
|
7
|
9.6
|
70
|
335
|
4.5
|
0.45
|
|
8
|
32.5
|
70
|
335
|
4.5
|
0.49
|
|
9
|
9.6
|
40
|
305
|
7.5
|
0.3
|
|
10
|
32.5
|
40
|
305
|
7.5
|
0.35
|
|
11
|
9.6
|
70
|
305
|
7.5
|
0.45
|
|
12
|
32.5
|
70
|
305
|
7.5
|
0.82
|
|
13
|
9.6
|
40
|
335
|
7.5
|
0.24
|
|
14
|
32.5
|
40
|
335
|
7.5
|
0.54
|
|
15
|
9.6
|
70
|
335
|
7.5
|
0.35
|
|
16
|
32.5
|
70
|
335
|
7.5
|
0.51
|
|
17
|
60
|
55
|
320
|
6
|
0.53
|
|
18
|
5
|
55
|
320
|
6
|
0.56
|
|
19
|
17.5
|
85
|
320
|
6
|
0.67
|
|
20
|
17.5
|
25
|
320
|
6
|
0.45
|
|
21
|
17.5
|
25
|
350
|
6
|
0.41
|
|
22
|
17.5
|
25
|
290
|
6
|
0.23
|
|
23
|
17.5
|
25
|
320
|
9
|
0.41
|
|
24
|
17.5
|
25
|
320
|
3
|
0.47
|
|
25
|
17.5
|
25
|
320
|
6
|
0.32
|
(a) Suppose you want to fit a second-order polynomial model to the data. Write the equations for least square regression in vector/matrix form. Define all the variables in your equation and specify the dimensions of each. Write down the matrix of independent variable values, X .
Hint: a second-order polynomial model can be written as
yˆ = β0 + β1x1 + β2x2 + β3x3 + β4x4 + β5x12 + ......+ β8x42 + ... + β9x1x2 + ... + β14x3x4.
(b) Make what you think is an appropriate analysis of the data to obtain your best model for radial run-out, y :
i. Describe your approach and steps of analysis.
ii. Report your model expression, the overall goodness-of-fit (e.g. ANOVA, esp. Rsquare and Adjusted Rsquare), estimated parameters, and their significance results (t-ratio or p-value).
Hint: use stepwise least square regression. The terms included in the final model should be either a significant variable or part of a significant variable.
(c) Based on your final model, make a two-dimensional contour plot of yˆ versus x1 and x3 over the ranges 5 ≤ x1 ≤ 60 and 290 ≤ x3 ≤ 350 , for each of the following three sets of conditions:
i. x2 = 40, x4 = 3
ii. x2 = 55, x2 = 55,
iii. x4 = 9 x4 = 4.5
Condition iii is the current condition used in the plant for production.
(d) Comment on this conclusion: "It was surprising to observe that either very wide (9-inch) or very narrow (3-inch) rims could be used to reach low radial run-out levels". Is it true? Use you model to explain why it is true/not true.
(e) Based on your model, how you would vary the factors studied above to achieve low radial run-out? Besides the model results, discuss the practical/operational considerations in determining how the factors should be varied.
Question 4: More than one response variables.
The coded factor settings and results from an experiment are shown below. The objective is to look for the settings to achieve a high yield and low filtration time.
|
Trial x x Yield
|
Field
|
Crystal
|
Filtration
|
|
|
1
|
2
|
(gs)
|
Color
|
Growth
|
Time
|
|
1
|
-
|
-
|
21.1
|
Blue
|
None
|
150
|
|
2
|
-
|
0
|
23.7
|
Blue
|
None
|
10
|
|
3
|
-
|
+
|
20.7
|
Red
|
None
|
8
|
|
4
|
0
|
-
|
21.1
|
Slightly red
|
None
|
35
|
|
5
|
0
|
0
|
24.1
|
Blue
|
Very slight
|
8
|
|
6
|
0
|
+
|
22.2
|
Unobserved
|
Slight
|
7
|
|
7
|
+
|
-
|
18.4
|
Slightly red
|
Slight
|
18
|
|
8
|
+
|
0
|
23.4
|
Red
|
Much
|
8
|
|
9
|
+
|
+
|
21.9
|
Very red
|
Much
|
10
|
Variable level
|
Variable
|
-
|
0
|
+
|
|
x1 = Condensation temperature (C)
|
90
|
100
|
110
|
|
3
x2 = Amount of B (cm )
|
24.4
|
29.3
|
34.2
|
(a) Analyze the data to obtain your best model:
i. Describe your approach and steps of analysis.
ii. Report your model expression, the overall goodness-of-fit (e.g. ANOVA, esp. Rsquare and Adjusted Rsquare), estimated parameters, and their significance results (t-ratio or p-value).
Hint: fit a second-order polynomial model by stepwise least square regression with each of the two responses. Take the logarithm of filtration time to fit a better model.
(b) Based on your model, draw contour diagrams for yield and filtration time.
(c) Find the settings of give x1 and x2 within the range enclosed by the experiment design levels that
i. The highest predicted yield, and
ii. The lowest predicted filtration time.
Hint: optimize each response separately. If you use an analytical method, you should check whether the true maximum/minimum condition is satisfied.
(d) Specify the optimal set of conditions for x1 and x2 that will simultaneously give high yield and low filtration time. At this set of conditions what field color and how much crystal growth would you most likely expect?
Analyzing experiments of two-level full factorial design using least square regression in JMP:
1. Set up the coded factor levels from DOE>Custom Design or DOE>Full Factorial Design and make the design table.
2. Enter the response values from the experiments the design table.
3. Run the script "Model" from the generated design table or click Analyze>Fit model.
4. Check and configure the model specification in the pop-up window and click "Run".
5. Note that an effect is half the value of the corresponding parameter estimated in least square regression.
6. In the model results window, red triangle next to "Prediction Profiler" > Interaction Profiler will make the interaction profiles appear.
Attachment:- Assignment.rar